In the current paper, we re-examine an interesting aspect of
GUP, which is the prediction that not only is there a minimum length, but
that all measurable lengths are quantized and discrete. Note that the former does not automatically imply the latter. A similar set of results follows for
area and volume measurements as well. It has been shown earlier that
this discreteness holds not only for flat background spacetimes, but also
in the presence of weak gravity.
Specifically, in Ref. [1]}, the authors have shown that the one-dimensional space confining an elementary particle must be discrete, implying all measurable lengths are quantized in units of a fundamental length scale, which can be the Planck length. Subsequently, in Ref. [2]}, this was extended for a relativistic particle confined in a rectangular as well as in a spherical box in one, two, and three dimensional space by solving the GUP-modified Klein Gordon and Dirac equations. It was shown that the length, area, and volume of the box are quantized in units of a fundamental length scale. Extending the results of flat spacetime to weakly curved spacetime, the authors of Ref. [3]} have shown that the quantization of lengths, areas, and volumes continue to hold in the weak gravitational regime. These results support the fundamentally discrete nature of space and a breakdown of the spacetime continuum picture near the Planck scale.
Here we will explore the extent to which the length quantization holds for strong gravity, i.e., in the background of spacetimes of high curvature. As a concrete example of a spacetime with strong gravity, we first consider the Schwarzschild BH metric in its high curvature region. Without exploiting any other BH property such as its horizon or singularity, we show that our result for the quantization of the length of the box continues to hold. Then, we obtain similar results for the case of Reissner-Nordström (RN) BH and for the case of the cosmological Friedmann-Lema\(\hat{{\rm i}}\) tre-Robertson-Walker (FLRW) spacetime.
In the above background spacetimes, we apply the aforementioned GUP, which is covariant under the non-linear Lorentz transformation.
Following our earlier works, our strategy would be to
solve the scalar field equation in the background of
the aforementioned spacetimes, impose “box boundary conditions” such that the field is confined to a finite region of space and show that such a
region can only be of quantized or discrete size.
In other words, the length measurements, which require the confinement of one or multiple particles (described by the scalar field) within a boundary, signifying the end points of the length to be measured, imply in turn that the measurable lengths are discrete as opposed to continuous. Deriving this in the context of strong gravity completes this program. Although discreteness of lengths, areas, and volumes have been derived from other approaches in the past, our approach here for the length quantization is relatively simpler and does rely neither on any particular approach of quantum gravity, nor does it require additional assumptions [4]}, [5]}, [6]}.
